Average Error: 1.6 → 1.6
Time: 19.2s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(z, \frac{1}{a - t}, \frac{-t}{a - t}\right), y, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(\mathsf{fma}\left(z, \frac{1}{a - t}, \frac{-t}{a - t}\right), y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r388653 = x;
        double r388654 = y;
        double r388655 = z;
        double r388656 = t;
        double r388657 = r388655 - r388656;
        double r388658 = a;
        double r388659 = r388658 - r388656;
        double r388660 = r388657 / r388659;
        double r388661 = r388654 * r388660;
        double r388662 = r388653 + r388661;
        return r388662;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r388663 = z;
        double r388664 = 1.0;
        double r388665 = a;
        double r388666 = t;
        double r388667 = r388665 - r388666;
        double r388668 = r388664 / r388667;
        double r388669 = -r388666;
        double r388670 = r388669 / r388667;
        double r388671 = fma(r388663, r388668, r388670);
        double r388672 = y;
        double r388673 = x;
        double r388674 = fma(r388671, r388672, r388673);
        return r388674;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.6
Target0.4
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.6

    \[x + y \cdot \frac{z - t}{a - t}\]

  2. Simplified1.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)}\]
  3. Using strategy rm

  4. Applied div-sub1.6

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a - t} - \frac{t}{a - t}}, y, x\right)\]
  5. Using strategy rm

  6. Applied div-inv1.6

    \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \frac{1}{a - t}} - \frac{t}{a - t}, y, x\right)\]

  7. Applied fma-neg1.6

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, \frac{1}{a - t}, -\frac{t}{a - t}\right)}, y, x\right)\]

  8. Simplified1.6

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \frac{1}{a - t}, \color{blue}{\frac{-t}{a - t}}\right), y, x\right)\]

  9. Final simplification1.6

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \frac{1}{a - t}, \frac{-t}{a - t}\right), y, x\right)\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

  (+ x (* y (/ (- z t) (- a t)))))